Fractional model of the chemical inductor

A multitude of materials and processes of a different nature frequently exhibit, in addition to the classical capacitive arcs, inductive loops in the complex impedance plane. It is a stable and physically robust response originated, from a mathematical perspective, by the introduction of a capacitive coupling in the slow relaxation variable. Nevertheless, the dynamical behavior of such systems shows, in reality, pronounced experimental deviations from the ideal inductive behavior. For this reason, the scientific community in general claims for the development of a novel theory that helps to phenomenologically interpret this pattern change, also encompassing the associated inherent physical complexity that capitalizes on the vast majority of real-world processes. Here, we present a generalization of the classical fast-slow models to naturally explain the anomalous dynamics observed experimentally in different types of measurement techniques, such as the inductance dispersion in impedance, the fractional relaxation processes with negative spike components in the transient responses, and the inverted current-voltage hysteresis. From numerical simulations, we analyze in detail the crossover dynamics from capacitive to inductive properties from the perspective of the constant phase element. Our work devises a useful theoretical framework that explains, in frequency and time domain, the coexistence of dispersive features and the transformation of the electrical behavior in terms of an anomalous bifractional crossover. Finally, we show the dominant role of the fractional-order dynamics in the appearance of inverted hysteresis commonly found in current-voltage curves. Although in literature, everything looks, in a certain manner, simple, the reality is that many real-world materials exhibit an intricate and complex nature that leads to anomalous processes evidenced by dispersive dynamics observed through different measurement techniques. This change of pattern analyzed here is appealing as a mathematical tool to interpret the physical phenomena of many familiar systems.